Rigid Dualizing Complexes via Differential Graded Algebras

Amnon Yekutieli, BGU

Abstract: Rigid dualizing complexes were introduced by Van den
Bergh in the context of noncommutative algebraic geometry, where
they proved to be extremely useful. The advantage of rigidity is
that it eliminates automorphisms, thus making dualizing complexes
unique, and even functorial.

This talk is about rigid dualizing complexes over *commutative*
K-algebras. If K is a field then the "noncommutative" results
specialize to yield an enormous amount of information on rigid
dualizing complexes and their variance properties. Indeed, one
can recover most of the important features of Grothendieck
duality (for affine schemes), including explicit formulas, with
relatively little effort.

However we want to consider commutative algebras over any
noetherian commutative base ring K. It turns out that this causes
serious technical issues, due to lack of flatness. Even defining
rigidity (i.e. writing Van den Bergh's rigidity equation) is a
problem! Our solution was to use differential graded algebras.
Thus, if A is a K-algebra which is not flat, we replace A with a
quasi-isomorphic DG K-algebra A^~  which has suitable flatness
properties, and use A^~ to formulate the rigidity equation for
complexes of A-modules.

Actually, this method enables us to work with relative rigid
complexes. Namely, given a homomorphism A -> B between
K-algebras, we can consider rigid complexes of B-modules relative
to A. (This is nontrivial even when the base K is a field.) The
theory of rigid complexes we developed is quite rich, and may be
of independent interest in ring theory. In this talk I will
explain some of the features of this theory.

When our base ring K is regular (e.g. the ring of integers) we
obtain a comprehensive theory of rigid dualizing complexes, once
again producing most of the important features of Grothendieck
duality for affine K-schemes. I'll discuss the main results, and
also some geometric consequences.

Full details can be found in the preprint math.AG/0601654 at
http://arxiv.org.

This is joint work with James Zhang (Univ. of Washington).